Finite Group Databasebeta

← C3×#25645 #10644252

Subgroups — C3×#25645 #10644252

Order 768.

1366 conjugacy classes of subgroups.

The subgroup lattice has 468 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C3×#550C211yesmaximal, characteristic
2C2×C3×#486C211yesmaximal, characteristic, complemented
3C2×C3×#474C211yesmaximal, characteristic, complemented
4C3×#2546C211yesmaximal, characteristic, complemented
5C3×#2041C214yesmaximal, complemented
6C3×#1984C211yesmaximal, characteristic, complemented
7C3×#1982C211yesmaximal, characteristic, complemented
8C3×#1862C214yesmaximal, complemented
9C3×C4×#102C211yesmaximal, characteristic, complemented
10#25645C311yesmaximal, characteristic, direct product factor, Sylow
11C22×C3×#41—21no—
12C3×#550—24no—
13C3×#550—24no—
14C2×C3×C4⋊Q8—21no—
15C2×C3×C4⋊Q8C2211yescharacteristic
16C2×C3×#127C2212yes—
17C2×C3×#127C2211yescharacteristic
18C2×C3×#121—21no—
19C2×C3×#121—22no—
20C2×C3×C4×Q8C2211yescharacteristic
21C2×C3×C4×D4C2212yes—
22C2×C3×C4×D4C2211yescharacteristic
23C3×#486—24no—
24C3×#474—24no—
25C2×C3×#107C2212yes—
26C2×C3×C4⋊C8C2211yescharacteristic
27C2×C3×#102C2214yes—
28C2×C3×#102C2212yes—
29C2×C3×#102C2212yes—
30C2×C3×#102C2214yes—
31C2×C3×C4×C8C2211yescharacteristic
32C3×#398—22no—
33C3×#391—21no—
34C3×#389C2214yes—
35C3×C4×C4⋊C4C2211yescharacteristic
36C3×#340C2212yes—
37C3×#336C2214yes—
38C2×#550C2×C311yescharacteristic
39C2×#486C2×C311yescharacteristic, complemented
40C2×#474C2×C311yescharacteristic, complemented
41#2546C2×C311yescharacteristic, complemented
42#2041C2×C314yescomplemented
43#1984C2×C311yescharacteristic, complemented
44#1982C2×C311yescharacteristic, complemented
45#1862C2×C314yescomplemented
46C4×#102C2×C311yescharacteristic, complemented
47C2×C3×#41—28no—
48C2×C3×#41—44no—
49C2×C3×#41—44no—
50C22×C3×Q8—21no—
51C22×C3×Q8D411yescharacteristic
52C22×C3×D4—22no—
53C22×C3×D4—21no—
54C22×C3×D4—22no—
55C22×C3×D4—21no—
56C22×C3×D4C2312yes—
57C22×C3×D4C2311yescharacteristic
58C23×C3×C4—22no—
59C23×C3×C4—21no—
60C22×C3×C8C2312yes—
61C22×C3×C8C2312yes—
62C3×C4⋊Q8—22no—
63C3×C4⋊Q8—22no—
64C3×C4⋊Q8—44no—
65C3×#127—24no—
66C3×#127—24no—
67C3×#127—22no—
68C3×#127D414yes—
69C3×#121—28no—
70C3×#121—216no—
71C3×C4×Q8—22no—
72C3×C4×Q8—22no—
73C3×C4×D4—24no—
74C3×C4×D4—24no—
75C3×C4×D4—22no—
76C3×C4×D4D414yes—
77C2×C3×C4⋊C4—24no—
78C2×C3×C4⋊C4—21no—
79C2×C3×C4⋊C4D411yescharacteristic
80C2×C3×C4⋊C4D412yes—
81C2×C3×C4⋊C4C2312yes—
82C2×C3×C4⋊C4C2311yescharacteristic
83C2×C3×C4⋊C4C2312yes—
84C2×C3×C22⋊C4—22no—
85C2×C3×C22⋊C4—21no—
86C2×C3×C42—21no—
87C2×C3×C42C2312yes—
88C2×C3×C42C2311yescharacteristic
89C3×#107—24no—
90C3×C4⋊C8—22no—
91C3×#102—24no—
92C3×#102—28no—
93C3×#102—24no—
94C3×#102—28no—
95C3×C4×C8—22no—
96C3×#95—21no—
97C3×#95—24no—
98C22×#41—21no—
99#550—24no—
100#550—24no—
101C2×C4⋊Q8—21no—
102C2×C4⋊Q8C22×C311yescharacteristic
103C2×#127C22×C312yes—
104C2×#127C22×C311yescharacteristic
105C2×#121—21no—
106C2×#121—22no—
107C2×C4×Q8C22×C311yescharacteristic
108C2×C4×D4C22×C312yes—
109C2×C4×D4C22×C311yescharacteristic
110#486—24no—
111#474—24no—
112C2×#107C22×C312yes—
113C2×C4⋊C8C22×C311yescharacteristic
114C2×#102C22×C314yes—
115C2×#102C22×C314yes—
116C2×#102C22×C312yes—
117C2×#102C22×C312yes—
118C2×C4×C8C22×C311yescharacteristic
119#398—22no—
120#391—21no—
121#389C22×C314yes—
122C4×C4⋊C4C22×C311yescharacteristic
123#340C22×C312yes—
124#336C22×C314yes—
125C24×C3—22no—
126C24×C3—21no—
127C3×#41—432no—
128C2×C3×Q8—22no—
129C2×C3×Q8—22no—
130C2×C3×Q8—42no—
131C2×C3×Q8—42no—
132C2×C3×Q8—42no—
133C2×C3×Q8D814yes—
134C2×C3×D4—28no—
135C2×C3×D4—28no—
136C2×C3×D4—24no—
137C2×C3×D4—24no—
138C2×C3×D4—22no—
139C2×C3×D4—21no—
140C2×C3×D4—22no—
141C2×C3×D4—24no—
142C2×C3×D4—22no—
143C2×C3×D4—24no—
144C2×C3×D4—44no—
145C2×C3×D4—44no—
146C2×C3×D4—42no—
147C2×C3×D4—42no—
148C2×C3×D4—82no—
149C2×C3×D4—84no—
150C2×C3×D4C2×D414yes—
151C2×C3×D4C2×D412yes—
152C22×C3×C4—28no—
153C22×C3×C4—22no—
154C22×C3×C4—21no—
155C22×C3×C4—24no—
156C22×C3×C4—22no—
157C22×C3×C4—44no—
158C22×C3×C4—44no—
159C22×C3×C4—42no—
160C22×C3×C4—42no—
161C22×C3×C4C2×D412yes—
162C22×C3×C4C2×D411yescharacteristic
163C22×C3×C4C2×D412yes—
164C22×C3×C4C2×D411yescharacteristic
165C22×C3×C4C2411yescharacteristic
166C2×C3×C8—22no—
167C2×C3×C8—22no—
168C2×C3×C8—24no—
169C2×C3×C8—22no—
170C2×C3×C8—22no—
171C3×C4⋊C4—24no—
172C3×C4⋊C4—22no—
173C3×C4⋊C4—21no—
174C3×C4⋊C4—22no—
175C3×C4⋊C4—22no—
176C3×C4⋊C4—22no—
177C3×C4⋊C4—42no—
178C3×C4⋊C4—48no—
179C3×C4⋊C4C2×D412yes—
180C3×C4⋊C4#4114yescomplemented
181C3×C22⋊C4—28no—
182C3×C22⋊C4—24no—
183C3×C42—22no—
184C3×C42—22no—
185C3×C42—42no—
186C3×C42C2×D412yes—
187C3×C42C2×D412yes—
188C2×#41—28no—
189C2×#41—44no—
190C2×#41—44no—
191C22×Q8—21no—
192C22×Q8C3×D411yescharacteristic
193C22×D4—22no—
194C22×D4—21no—
195C22×D4—22no—
196C22×D4—21no—
197C22×D4C23×C312yes—
198C22×D4C23×C311yescharacteristic
199C23×C4—22no—
200C23×C4—21no—
201C22×C8C23×C312yes—
202C22×C8C23×C312yes—
203C4⋊Q8—22no—
204C4⋊Q8—22no—
205C4⋊Q8—44no—
206#127—24no—
207#127—24no—
208#127—22no—
209#127C3×D414yes—
210#121—28no—
211#121—216no—
212C4×Q8—22no—
213C4×Q8—22no—
214C4×D4—24no—
215C4×D4—24no—
216C4×D4—22no—
217C4×D4C3×D414yes—
218C2×C4⋊C4—24no—
219C2×C4⋊C4—21no—
220C2×C4⋊C4C3×D411yescharacteristic
221C2×C4⋊C4C3×D412yes—
222C2×C4⋊C4C23×C312yes—
223C2×C4⋊C4C23×C311yescharacteristic
224C2×C4⋊C4C23×C312yes—
225C2×C22⋊C4—22no—
226C2×C22⋊C4—21no—
227C2×C42—21no—
228C2×C42C23×C312yes—
229C2×C42C23×C311yescharacteristic
230#107—24no—
231C4⋊C8—22no—
232#102—24no—
233#102—28no—
234#102—28no—
235#102—24no—
236C4×C8—22no—
237#95—21no—
238#95—24no—
239C23×C3—24no—
240C23×C3—22no—
241C23×C3—42no—
242C23×C3—44no—
243C23×C3—42no—
244C23×C3—41no—
245C23×C3—42no—
246C23×C3—41no—
247C23×C3—44no—
248C23×C3—42no—
249C23×C3C22×D411yescharacteristic
250C3×Q8—28no—
251C3×Q8—84no—
252C3×D4—22no—
253C3×D4—22no—
254C3×D4—28no—
255C3×D4—416no—
256C3×D4—42no—
257C3×D4—42no—
258C3×D4—42no—
259C3×D4—48no—
260C3×D4—84no—
261C3×D4—88no—
262C2×C3×C4—22no—
263C2×C3×C4—22no—
264C2×C3×C4—22no—
265C2×C3×C4—22no—
266C2×C3×C4—22no—
267C2×C3×C4—24no—
268C2×C3×C4—24no—
269C2×C3×C4—416no—
270C2×C3×C4—42no—
271C2×C3×C4—41no—
272C2×C3×C4—41no—
273C2×C3×C4—42no—
274C2×C3×C4—48no—
275C2×C3×C4—42no—
276C2×C3×C4—42no—
277C2×C3×C4—42no—
278C2×C3×C4#12012yes—
279C2×C3×C4#12012yes—
280C2×C3×C4C2×D814yes—
281C2×C3×C4C2×D812yes—
282C2×C3×C4AGL(1,ℤ/8)12yes—
283C2×C3×C4C22×D411yescharacteristic
284C2×C3×C4C22×D411yescharacteristic
285C2×C3×C4C2×#4112yes—
286C2×C3×C42−1+412yes—
287C3×C8—42no—
288C3×C8—42no—
289C24—22no—
290C24—21no—
291#41—432no—
292C2×Q8—22no—
293C2×Q8—22no—
294C2×Q8—42no—
295C2×Q8—42no—
296C2×Q8—42no—
297C2×Q8C3×D814yes—
298C2×D4—28no—
299C2×D4—28no—
300C2×D4—24no—
301C2×D4—24no—
302C2×D4—22no—
303C2×D4—21no—
304C2×D4—22no—
305C2×D4—24no—
306C2×D4—22no—
307C2×D4—24no—
308C2×D4—44no—
309C2×D4—44no—
310C2×D4—42no—
311C2×D4—42no—
312C2×D4—82no—
313C2×D4—84no—
314C2×D4C2×C3×D414yes—
315C2×D4C2×C3×D412yes—
316C22×C4—28no—
317C22×C4—22no—
318C22×C4—21no—
319C22×C4—24no—
320C22×C4—22no—
321C22×C4—44no—
322C22×C4—44no—
323C22×C4—42no—
324C22×C4—42no—
325C22×C4C2×C3×D412yes—
326C22×C4C2×C3×D411yescharacteristic
327C22×C4C2×C3×D412yes—
328C22×C4C2×C3×D411yescharacteristic
329C22×C4C24×C311yescharacteristic
330C2×C8—22no—
331C2×C8—22no—
332C2×C8—24no—
333C2×C8—22no—
334C2×C8—22no—
335C4⋊C4—24no—
336C4⋊C4—22no—
337C4⋊C4—21no—
338C4⋊C4—22no—
339C4⋊C4—22no—
340C4⋊C4—22no—
341C4⋊C4—42no—
342C4⋊C4—48no—
343C4⋊C4C2×C3×D412yes—
344C4⋊C4C3×#4114yescomplemented
345C22⋊C4—28no—
346C22⋊C4—24no—
347C42—22no—
348C42—22no—
349C42—42no—
350C42C2×C3×D412yes—
351C42C2×C3×D412yes—
352C22×C3—42no—
353C22×C3—44no—
354C22×C3—44no—
355C22×C3—42no—
356C22×C3—44no—
357C22×C3—42no—
358C22×C3—82no—
359C22×C3—84no—
360C22×C3—84no—
361C22×C3—82no—
362C22×C3C2×#12011yescharacteristic
363C22×C3#54812yes—
364C22×C3C22×D811yescharacteristic
365C22×C3C2×AGL(1,ℤ/8)11yescharacteristic
366C22×C3#57712yes—
367C3×C4—24no—
368C3×C4—22no—
369C3×C4—44no—
370C3×C4—42no—
371C3×C4—42no—
372C3×C4—82no—
373C3×C4#44714yes—
374C3×C4#53211yescharacteristic
375C3×C4#53611yescharacteristic
376C3×C4#54712yes—
377C23—24no—
378C23—22no—
379C23—42no—
380C23—41no—
381C23—42no—
382C23—41no—
383C23—44no—
384C23—42no—
385C23—42no—
386C23—44no—
387C23C22×C3×D411yescharacteristic
388Q8—28no—
389Q8—84no—
390D4—22no—
391D4—22no—
392D4—28no—
393D4—416no—
394D4—48no—
395D4—42no—
396D4—42no—
397D4—42no—
398D4—84no—
399D4—88no—
400C2×C4—22no—
401C2×C4—22no—
402C2×C4—22no—
403C2×C4—24no—
404C2×C4—22no—
405C2×C4—22no—
406C2×C4—24no—
407C2×C4—416no—
408C2×C4—42no—
409C2×C4—41no—
410C2×C4—41no—
411C2×C4—42no—
412C2×C4—48no—
413C2×C4—42no—
414C2×C4—42no—
415C2×C4—42no—
416C2×C4C3×#12012yes—
417C2×C4C3×#12012yes—
418C2×C4C2×C3×D814yes—
419C2×C4C2×C3×D812yes—
420C2×C4C3×AGL(1,ℤ/8)12yes—
421C2×C4C22×C3×D411yescharacteristic
422C2×C4C22×C3×D411yescharacteristic
423C2×C4C2×C3×#4112yes—
424C2×C4C3×2−1+412yes—
425C8—42no—
426C8—42no—
427C2×C3—82no—
428C2×C3—84no—
429C2×C3#243211yescharacteristic
430C2×C3C2×#44711yescharacteristic
431C2×C3#300911yescharacteristic
432C2×C3#335812yes—
433C2×C3#336812yes—
434C22—42no—
435C22—44no—
436C22—44no—
437C22—42no—
438C22—44no—
439C22—42no—
440C22—82no—
441C22—84no—
442C22—84no—
443C22—82no—
444C22C2×C3×#12011yescharacteristic
445C22C3×#54812yes—
446C22C22×C3×D811yescharacteristic
447C22C2×C3×AGL(1,ℤ/8)11yescharacteristic
448C22C3×#57712yes—
449C4—24no—
450C4—22no—
451C4—42no—
452C4—42no—
453C4—44no—
454C4—82no—
455C4C3×#44714yes—
456C4C3×#53211yescharacteristic
457C4C3×#53611yescharacteristic
458C4C3×#54712yes—
459C3#2564511yescharacteristic, direct product factor, Sylow
460C2—82no—
461C2—84no—
462C2C3×#243211yescharacteristic
463C2C2×C3×#44711yescharacteristic
464C2C3×#300911yescharacteristic
465C2C3×#335812yes—
466C2C3×#336812yes—
subgroupG/H
center, socleC23×C3C22×D4
commutator, Frattini, cosocleC22×C4C24×C3
prime radicalC2×C3×#550C2
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23×C3 ◁ C2×C3×C4×Q8 ◁ G

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
2. C2×C3×#486C2semidirect
3. C2×C3×#474C2semidirect
4. C3×#2546C2semidirect
5. C3×#2041C2semidirect
6. C3×#1984C2semidirect
7. C3×#1982C2semidirect
8. C3×#1862C2semidirect
9. C3×C4×#102C2semidirect
10. #25645C3direct (N × H)
39. C2×#486C2×C3semidirect
40. C2×#474C2×C3semidirect
41. #2546C2×C3semidirect
42. #2041C2×C3semidirect
43. #1984C2×C3semidirect
44. #1982C2×C3semidirect
45. #1862C2×C3semidirect
46. C4×#102C2×C3semidirect
180. C3×C4⋊C4#41semidirect
344. C4⋊C4C3×#41semidirect
459. C3#25645direct (N × H)

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroupindex
C2×C3×#5502
C2×C3×#4862
C2×C3×#4742
C3×#25462
C3×#20412
C3×#19842
C3×#19822
C3×#18622
C3×C4×#1022
#256453